Tracking and characterizing particles with holographic video microscopy

ABSTRACT

In-line holography to create images of a specimen, such as one or more particles dispersed in a transparent medium. Analyzing these images with results from light scattering theory yields the particles&#39; sizes with nanometer resolution, their refractive indexes to within one part in a thousand, and their three dimensional positions with nanometer resolution. This procedure can rapidly and directly characterize mechanical, optical and chemical properties of the specimen and its medium.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a continuation of U.S. application Ser. No.12/740,628 filed Nov. 12, 2010, which is a national stage application ofPCT Patent Application No. PCT/US2008/081794, filed Oct. 30, 2008, andclaims priority from the benefit under 35 USC 119(e) of U.S. ProvisionalPatent Application 61/001,023, filed Oct. 30, 2007 and U.S. ProvisionalPatent Application 61/073,959, filed Jun. 19, 2008, incorporated hereinby reference in their entirety.

This work was supported by the National Science Foundation under GrantNumber DMR-0606415.

BACKGROUND OF THE INVENTION

The present system and method are directed to measuring thethree-dimensional position of colloidal particles within colloidaldispersions, and furthermore to characterizing each particle's shape,size, and optical properties, by quantitative analysis of hologramsrecorded through holographic video microscopy. Information obtained fromthis analysis can be used to characterize the colloidal particlesthemselves. This information also can be used to measure and assay theproperties of the medium within which the particles are dispersed. Inall such applications, quantitative holographic video microscopy offerssubstantial advantages over methods and analysis systems that have beenpreviously described and practiced.

Particle-image velocimetry is widely applied for measuring thetrajectories of colloidal particles dispersed in transparent media.Conventional implementations of particle-imaging velocimetry involveforming images of the particles using standard light microscopy and thenanalyzing those images through methods of computer image analysis. Suchmeasurements and analyses typically provide information on individualparticles' positions in the microscope's focal plane, but not on theparticles' axial displacements relative to the focal plane. In caseswhere axial tracking information is extracted, the results typicallyhave substantially poorer spatial resolution than the in-planepositions. Measurements of axial positions, furthermore, requireseparate calibration measurements for each particle. Such conventionalparticle-imaging methods generally provide little information on theparticles' sizes, shapes, or compositions. The range of axialdisplacements over which conventional particle-imaging methods can beapplied is limited by the depth of focus of the microscope becauseimages of particles that move too far from the focal plane become toodim and diffuse to analyze.

Applications of image-based particle tracking include measuringstreamlines in flowing fluids, assessing the thermodynamic stability ofcolloidal dispersions against aggregation and flocculation, measuringinteractions among colloidal particles, measuring colloidal particles'interactions with surfaces, assessing particles' responses to externalfields and forces, characterizing the particles' viscous dragcharacteristics, and using the particles' motions as probes of theviscoelastic and rheological properties of the embedding medium. Thelatter class of measurements, in which colloidal particles are used asmicroscopic probes of the medium's rheology, is commonly termedparticle-tracking microrheology. All such applications benefit fromparticle-tracking techniques that offer better spatial resolution,three-dimensional tracking, and a wider axial range. Some of theseapplications, such as microrheology, also require information on theprobe particles' characteristics, such as their radii. Typically, suchparticle characterization data are obtained in separate measurements.

In the particular case of microrheology, other methods are available foracquiring equivalent information on a medium's viscoelastic properties.Among these are diffusing wave spectroscopy, dynamic light scatteringand interferometric particle tracking. All such methods offer superiorbandwidth to those based on particle imaging. The first two do not,however, offer spatially resolved measurements, which are necessary insome applications. Interferometric particle tracking offers bothexcellent bandwidth and excellent tracking resolution. It can only beapplied to one or two points in a sample, however, and so cannot be usedfor multi-point assays of rheological properties. None of these methodsis suitable for analyzing the properties of inhomogeneous samples.

Individual colloidal particles typically are characterized by theirshape, their size, their bulk composition, and their surface properties.Colloidal dispersions are characterized by the distributions of thesequantities as well as by the overall concentration of particles. Sizeand shape can be assessed through electron microscopy on dried andotherwise prepared samples. Preparation can change the particles'properties, however, so that the results of such measurements might notaccurately reflect the particles' characteristics in situ. Lightscattering methods generally are used for in situ analysis of colloidalparticles' sizes. Such measurements, however, provide a sample-averagedview of the particles in a dispersion, and generally require carefulinterpretation with approximate or phenomenological models for the theparticles' size distribution, shapes, and refractive indexes. Commonlyused commercial particle sizing instruments are based on these methodsand share their limitations. These methods, furthermore, cannot be usedto characterize the particular particles used in particle-trackingmeasurements. Other particle-sizing instruments, such as Coultercounters, similarly rely on indirect methods to measure particle sizesand cannot be applied in situ. A variety of methods also are known frommeasuring colloidal particles' refractive indexes. Conventional lightscattering methods generally provide sample-averaged values for therefractive index and require information on the particles' sizes andshapes. A particularly effective method involves matching the refractiveindex of a fluid to that of the particles and then measuring therefractive index of the fluid. This method requires index-matchingfluids that are compatible with the colloids and is highly limited inthe range of refractive indexes that can be assessed.

SUMMARY OF THE INVENTION

Quantitative analysis of images obtained through holographic videomicroscopy provides the information required to simultaneously track andcharacterize large numbers of colloidal particles, and yieldsinformation on each particle individually. For tracking particles, thismethod offers nanometer-scale position measurements in three dimensionsover an exceedingly large range of axial positions. Such high-resolutionwide-range three-dimensional tracking data is ideal for applications inmicrorheology, and similarly can be used beneficially in any applicationwhere conventional particle tracking currently is applied. Holographicparticle tracking works for particles ranging in size from a fewnanometers up to at least a few micrometers. It offers the full timeresolution of the video camera used to acquire the holograms. Forparticle characterization, holographic analysis yields the radius withnanometer resolution and the complex refractive index with a relativeerror no worse than one part in a thousand. It covers the full range ofrefractive indexes from low-index bubbles in high-index media toparticles whose refractive indexes are too high to be assessed by othermeans. These specifications surpass those of many methods devoted toparticle characterization alone. The combination of high-resolutionparticle tracking with precise in situ particle characterization makespossible measurements that cannot be performed by other means.

Various aspects of the invention are described herein; and these andother improvements are described in greater detail hereinbelow,including the drawings described hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a schematic representation of a holographic videomicroscope; FIG. 1B shows the resulting light scattering in an imagingplane depicted in FIG. 1A; FIG. 1C illustrates a resulting measuredhologram and FIG. 1D shows a fitted hologram using a method of theinvention;

FIG. 2A illustrates a schematic flow diagram of a preferred embodimentof analyzing a specimen to provide characteristic property information;FIG. 2B shows details of a computer software driven methodology ofprocessing the specimen data;

FIG. 3A illustrates a 3-D plot of a trajectory of a polystyrene beadfreely diffusing in a Newtonian fluid; and FIG. 3B shows mean-squaredisplacement for each x, y, z coordinate as a function of time;

FIG. 4A shows a measure of the viscoelastic moduli, G′(ω) and G″(ω),extracted from the data of FIGS. 3A and 3B; FIG. 4B shows associateddynamic viscosity, η(ω); FIG. 4C shows viscoelastic moduli for a 17 wt.% sample of 250k Da polyethylene oxide (“PEO”) in water with the insetgraph showing mean square displacement of the probe particle'strajectory; and FIG. 4D shows the dynamic viscosity;

FIG. 5A shows viscoelastic moduli, G′(ω) and G″(ω), of a reconstitutedS-type polysaccharide biofilm with a graph inset of the mean squaredisplacement; FIG. 5B shows dynamic viscosity of the S-type biofilm;FIG. 5C shows the same variables as FIG. 5A for an N-type polysaccharideat same concentration and FIG. 5D shows the same variables as FIG. 5Bfor the same concentration for an N-type polysaccharide;

FIG. 6A illustrates fat globule radius versus refractive index forcommercial milk samples, each data point representing results for asingle fat droplet (error bars are computed from the normalized varianceindex of fitting parameters); FIG. 6B shows count distribution versusdiameter and FIG. 6C shows count distribution versus refractive index;

FIG. 7A shows refractive index versus droplet radius for dispersed typeB immersion oil in water, with and without a surfactant; FIG. 7B showscount distribution versus diameter and FIG. 7C shows count distributionversus refractive index;

FIG. 8A(1) shows a normalized hologram B(ρ) from a polystyrene sphere of1.43 μm diameter in water at z_(p)=22.7 μm; FIG. 8A(2) is a numericalfit using Eq. (18); FIG. 8A(3) shows the azimuthally averaged radialprofile B(ρ) for the sphere; FIG. 8B(1) shows hologram data for a 1.45μm diameter TiO₂ sphere dispersed in immersion oil (n_(m)=1.515) atz_(p)=7.0 μm; FIG. 8B(2) shows the numerical fit as in the method ofFIG. 8A(3); FIG. 8B(3) shows the corresponding azimuthally averagedradial profile B(ρ); FIG. 8C(1) shows hologram data for a 4.5 μm SiO₂sphere in water at z_(p)=38.8 μm; FIG. 8C(2) shows the numerical fit asin the method of FIG. 8A(3) and FIG. 8C(3) shows the correspondingazimuthally averaged radial profile B(ρ);

FIG. 9A(1) shows digital hologram images of a colloidal silica sphere atthe beginning of a trajectory; FIG. 9A(2) shows the colloidal silicasphere at the end of the trajectory; FIG. 9B(1) shows the correspondingnumerical fit using Eq. (18) at the beginning of the trajectory and FIG.9B(2) the corresponding fit at the end of the trajectory;

FIG. 10 illustrates a 3-D trajectory of the colloidal silica sphereshowing its starting point (a circle) and end point (square) labeled;

FIG. 11 shows the function z(t) thermal fluctuation associated withthermal sedimentation (inset graph is a fit of the refractive indexwhich is independent of position); and

FIG. 12 shows mean square positional fluctuations as a function of τ(sec.) with Einstein-Smoluchowsky x, y, z scaling and with an inset plotof the x, y plane of trajectory projections.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The subject holographic microscope 10, depicted schematically in FIG.1A, is based on a commercial inverted light microscope (Zeiss AxiovertTV 100 S). A conventional incandescent illuminator is replaced with acollimated 10 mW HeNe laser 20 (for example, Uniphase) operating at avacuum wavelength of λ=632.8 nm. A specimen, such as an individualparticle 12 or different phase 30 within a medium 35 disposed on aspecimen stage or holder 38 at position r_(p) scatters a small portionof the plane-wave illumination light 40. Scattered light 50 thenpropagates to the focal plane 60 of the microscope 10, where the light50 interferes with the unscattered portion of the laser beam light 40.The resulting interference pattern is magnified by the microscope'sobjective lens 70 (such as, an S Plan Apo, 100×, NA 1.4, oil immersion)and projected by a video eyepiece 80 (0.63×) onto a CCD camera 15 (suchas, NEC TI-324AII) before being recorded as uncompressed digital videowith a digital video recorder 18 (for example, Panasonic DVR-H110, notshown) and accumulated data is processed by a computer 100 executingvarious computer software embedded in module 120. In a preferred form ofthe invention the computer software on the module 120 includes a set ofinstructions for operating on data characteristic of the specimen. Thedata characteristic of the specimen 12, 30 relates to an interferencepattern arising from interaction of the laser light 50 scattered fromthe specimen 12, 30 and an unscattered portion of the laser light 40.The set of instructions further include a scattering function which uponexecution provides a convergent solution descriptive of properties ofthe specimen 12, 30, thereby enabling characterization of at least oneof a mechanical property, an optical property, and a chemical propertyof the specimen 12, 30.

Each holographic image in the video stream provided by the recorder 18is a time-resolved snapshot of the three-dimensional distribution ofscatterers in the microscope's field of view. In additional embodimentsa plurality of time snapshots allow tracking particle trajectory andobtaining characteristic properties. We then use results of theLorenz-Mie theory of light scattering (or other appropriate lightscattering methodologies embedded as a program in the software module120) by the small particles 12 to measure characteristics of eachparticle 12 or the different phase 30. In the system of FIG. 1A, thespecimen data output from the recorder 18 and processed by the computer100 can be output to a control system 150 for generating quality controlor manufacturing control signals 100 for the specimen 12, 30.

In a most preferred embodiment the microscope 10 is operated inaccordance with the flow diagram shown in FIGS. 2A and 2B. Acquisitionof holographic images begins with the choice of the laser 20 used toilluminate the specimen 12, 30. Characteristics of the laser 20 that maybe selected for a particular application include the wavelength of thelight 40, the number of distinct wavelengths of the light 40, thecoherence length, the intensity of the light 40 at each wavelength, andwhether the laser 20 is continuous or pulsed. If the laser 20 is pulsed,the shape and timing of the pulses may be optimized for differentapplication. These choices typically would be made in consideration ofthe intended application and of the other hardware and softwarecomponents in the microscope 10. For example, a pulsed form of the laser20 might be preferable for time-resolved imaging of rapidly movingobjects. The pulses might then be synchronized with the shutter of thevideo recorder 18 or other camera used to record images. The wavelengthsimilarly might be selected to take advantage of optical properties ofthe specimen 12, 30. A plurality of wavelengths of the laser 20 might beselected to acquire spectroscopic information through analysis ofholograms recorded in different wavelengths. Holograms at differentwavelengths might be recorded simultaneously with a camera (such as thevideo recorder 18) capable of recording color information. They might berecorded with multiple cameras, each selected for a particularwavelength. Or, they might be recorded sequentially with one or morecameras.

For in-line holography, the specimen 12, 30 is most preferablysufficiently transparent for a sufficient fraction of the illuminatinglaser 20 to pass through undistorted so as to form an interferencepattern. The specimen 12, 30 thus has to be mounted in a transparentform of the sample holder 38; and the sample holder 38 mounted rigidlyin the laser beam 20, disposed so that the resulting interferencepattern is visible to the microscope 10. If these conditions are met,the light 40 scattered by the specimen 12, 30 to provide the light 50which will interfere with the unscattered portion of the laser beam 20to form an interference pattern, or hologram. The role of the microscope10 is to magnify the hologram and project the magnified hologram ontothe camera 18. The microscope's objective lens 70 and eyepiece 80 thusis preferably selected to optimize collection, magnification andprojection of the light 40, at the wavelength of choice.

The video recorder or camera 18, perhaps coupled with a separatedigitization system, records the hologram, digitizes it, and transfersthe digitized image to a digital image storage system (shown, forexample, as 25 in FIG. 1A). The camera's pixel count, spatial resolutionand frame rate establishes the spatial range, spatial resolution andtemporal bandwidth of measurements that can be performed withholographic microscopy methods. In addition, its dynamic rangeestablishes the intensity of the laser 20 required to acquire usefulimages and the amount of information that can be acquired from eachimage. Its ability to record color information determines what role thecamera 18 can play in acquiring holograms at multiple wavelengths.Images acquired by the camera 18 can be recorded for later analysis, ortransferred directly to image analysis routines for real-time analysis.

The images can be prepared for analysis and subsequent use in the mannershown in FIG. 2B. Images acquired by the camera 18 and, perhaps, datastored in the digital image storage system 25, consist of arrays ofpixels, each of which records the local intensity value as a number.Before these numbers can be analyzed with light scattering theory, theyare most preferably appropriately normalized. Normalization may consistof dividing the image by a previously recorded background image, asdescribed herein. This has the benefit of eliminating intensityvariations in the image due to spurious interference fringes.Alternatively, it may consist of dividing the entire image by anormalization constant, or normalizing the image by a numerical modelfor a background illumination pattern.

Once the image is normalized, it can be analyzed by fitting topredictions based on selected appropriate light scattering theory. Lightscattering theory comprises several different mathematical formulations,including Lorenz-Mie theory, T-matrix theory, and other approaches topredicting the electric and magnetic waves scattered by an illuminatedobject. Any such formulation may be used to analyze a normalized image.In the particular case that the specimen 12, 30 is a sphere, Lorenz-Mietheory is particularly well suited. For more complex structures,T-matrix theory may be preferable.

Once the appropriate formulation for the scattered light field has beenselected, the normalized image may be fit to the theory by manipulatingthe adjustable parameters. In the particular case of a homogeneousisotropic spherical object for the specimen 12, 30, the adjustableparameters include the sphere's three-dimensional position, its radius,and its complex refractive index. In the case of more complex objects,additional fitting parameters may be required. In general, the necessaryfit involves a highly nonlinear optimization of the adjustableparameters. In our reduction to practice of this method, we adopted theLevenberg-Marquardt nonlinear least-squares optimization algorithm.Other algorithms may be preferable for some applications. The fits maybe performed on a pixel-by-pixel basis using the CPU of the computer100. Alternatively, the computations may be accelerated though the useof a graphics processing unit (GPU), or another such parallel-processingsystem (also represented as “computer 100” in FIG. 1A).

Fits may be performed for the specimen 12, 30 in each recordedholographic image. Alternatively, some particular specimen 12, 30 may beselected in each image for further consideration. Once an image has beenanalyzed, the object data may be analyzed either in isolation or inconjunction with data from other images. The object data may be thedesired final product for use thereof. This could be the case forverifying a three-dimensional configuration of the specimen 12, 30, orassessing the optical characteristics of a sample of the specimen 12,30.

A sequence of holographic images of an individual one of the specimen12, 30 may be combined to develop a time-resolved measurement of thatspecimen's three-dimensional trajectory. Such trajectories themselvesmay be used as inputs for further analysis. For example, statisticalanalysis of trajectory data may be used to develop measurements of thevelocity field in a flowing fluid. Such measurements are conventionallyknown by the term particle-image velocimetry. The variation based onholographic particle tracking might therefore be termed holographicparticle-image velocimetry. Alternatively, trajectory data may beanalyzed to obtain information about the viscoelastic properties of themedium 35 within which a plurality of the specimen particles 18 isdispersed, which can be termed as holographic microrheology (this willbe described in more detail hereinafter in Section A).

As described in more detail hereinafter the specimen's size andrefractive index data and an individual particle's size and refractiveindex may be monitored over time. These values should remain constant ina stable system. Variations in these values provide information aboutthe changing conditions within the system. Such information may beuseful in such applications as holographic microrheology. Alternatively,the size and refractive index of a statistical ensemble of the particles18 or the phase 30 may be obtained from a plurality of holographicimages. Such ensembles may be analyzed to estimate the statisticaldistribution of sizes and refractive indexes within a bulk specimen 12,30. Such information is useful in monitoring processes and products forquality and consistency. In the case of non-spherical specimens 12, 30,additional information such as shape and orientation also may beobtained, and statistical analyses performed.

In some cases, the chemical composition of the specimen 12, 30 may beassessed by measuring its refractive index at a plurality ofwavelengths. In the case of heterogeneous specimen such as coatedspheres, composition analysis can detect the presence or composition ofcoatings on the surface of the specimen 12, 30. Composition analysis isuseful for process control and product quality assurance, for instanceas a means of detecting adulterants in emulsions or particle-basedproducts. Detection of molecular scale coatings is useful for label-freemolecular binding assays. Other applications of the geometric andcompositional data also should be apparent, including applications thatalso make use of tracking data.

A. Microrheology of a Medium

In various embodiments, this methodology was applied to analysis ofnumerous different types of the particles and phases 12, 30. In onepreferred embodiment we have performed holographic microrheologicalanalyses of a medium's viscoelastic properties by analyzing itsthermally-driven fluctuations, either directly, or through theirinfluence on embedded probe particles. In one embodiment the approachcan monitor a probe particle's mean-square positional fluctuations,

Δr²(t)

, and uses a generalized Stokes-Einstein relation to extract themedium's frequency-dependent storage modulus, G′(ω). This is related bythe Kramer-Kronig relation to the frequency-dependent loss modulus,G″(ω), which completes the micromechanical description.

Our example probe particles are charge-stabilized polystyrene spheres ofnominal radius a=0.75 μm (Duke Scientific, Catalog #5153A, Lot #26621)that are dispersed at random into a sample medium by vortexing. Thesample medium then is charged into a transparent container formed bybonding a no. 1 glass cover slip to the face of a microscope slide. Thesealed sample medium is allowed to come to thermal and mechanicalequilibrium on the microscope stage at T=23±1° C.

Holographic images are recorded as an uncompressed digital video streamat 30 frames per second on a digital video recorder (PanasonicDMR-E100H). Each image then is analyzed to measure the probe particle'sthree-dimensional location relative to the center of the microscope'sfocal plane.

More specifically, the collimated laser illuminates the probe particleat position r_(p) with a plane-wave incident electric field,E₀(r_(p))=u₀(r_(p))exp(−ikz_(p)), where k=2πn_(m)/λ is the wavenumber ofthe light in a medium of refractive index n_(m). The field scattered bythe particle, E_(s)(r)=u₀(r_(p))f_(m)(r−r_(p)), propagates to the focalplane at z=0, where it interferes with the incident beam. Thedistribution of scattered light is described by the conventionalLorenz-Mie scattering function, f_(m) (r−r_(p)), which depends on theparticle's position r_(p), its radius, a, and its refractive index,n_(p).

In practice, the incident illumination varies with position, so that wenormalize the measured interference pattern, I(r), by a measurement ofthe incident illumination I₀ (r)=|u₀(r)|² to obtain,

$\begin{matrix}{{\frac{I(r)}{I_{0}(r)} = {1 + {2{\alpha\Re}\left\{ {{{f_{s}\left( {r - r_{p}} \right)} \cdot \hat{ɛ}}e^{{- i}\;{kz}_{p}}} \right\}} + {\alpha^{2}{{f_{s}\left( {r - r_{p}} \right)}}^{2}}}},} & (1)\end{matrix}$where α≈1 accounts for variations in I₀(r_(p)). Equation (1) can be fitto normalized images such as the example shown in FIG. 1B, with theparticle's position, radius, and refractive index as free parameters.Whereas conventional bright-field particle tracking on the sameinstrument offers 10 nm in-plane resolution and 100 nm axial resolution,holographic particle tracking performs at least one order of magnitudebetter. Unlike conventional particle tracking, moreover, holographictracking does not require separate calibrations for axial measurements.

We can assess the measurement error in the particles' positions bytracking probe particles freely diffusing in Newtonian fluids such aswater. Provided the particle is far enough from bounding surfaces, itsmean-square displacement in each of the three Cartesian directionsshould evolve according to the Stokes-Einstein relation

Δr _(j) ²(t)

≡

|r _(j)(t+τ)−r _(j)(τ)|²

=2D ₀ t,  (2)Where D₀=k_(B)T/(6πηα) is the diffusion coefficient for a sphere in afluid of viscosity η at absolute temperature T. The angle brackets inEq. (2) denote an ensemble average over starting times. Restricting theaverage to starting times that are separated by the interval t ensuresthat contributions to

Δr_(j) ²(t)

are statistically independent. When analyzing a single discretelysampled trajectory, however, this choice yields disproportionately largestatistical errors at longer lag times, t. Averaging over all startingtimes improves the estimate for to

Δr_(j) ²(t)

and is justified if the trajectory may be treated as a Markov process.This is the case for the thermally driven trajectories we consider, andsuch exhaustive sampling enables us to estimate the mean-squareddisplacement from a single trajectory measured over a few thousand timesteps. Statistical errors in to

Δr_(j) ²(t)

must be corrected for covariances among correlated measurements over theinterval t.

Measurements of r_(j)(t) also suffer from random errors whose meanvalue, ∈_(j), establishes the tracking resolution. These errors increasethe particle's apparent mean-square displacement by 2∈_(j) ²,independent of t. A complementary error due to motional blurring duringthe camera's shutter period, τ_(s), reduces the apparent mean-squaredisplacement. The result is given by,

Δr _(j) ²(t)

=2D ₀ t+2(∈_(j) ²−⅓D ₀τ_(s))  (3)accounts for both effects, and enables us to measure ∈_(j).

The Fourier transform of to

Δr_(j) ²(t)

is related to the complex frequency-dependent viscoelastic modulusthrough the phenomenological generalized Stokes-Einstein relation,

$\begin{matrix}{{G^{*}(\omega)} = {{- i}\frac{k_{B}T}{\pi\;{\alpha\omega}\left\langle {\Delta\;{{\hat{r}}_{j}^{2}(\omega)}} \right\rangle}}} & (4) \\{\mspace{65mu}{\approx {i^{\alpha\;{j{(\omega)}}}\frac{k_{B}T}{\pi\;\alpha\left\langle {\Delta\;{r_{j}^{2}\left( {1/\omega} \right)}} \right\rangle{\Gamma\left( {1 + {\alpha_{j}(\omega)}} \right)}}}}} & (5)\end{matrix}$where Γ(x) is the gamma function and,

$\begin{matrix}{{\alpha_{j}(\omega)} = {{\frac{d\;\ln\left\langle {\Delta\;{r_{j}^{2}(t)}} \right\rangle}{d\;\ln\mspace{11mu} t}\text{|}t} = \frac{1}{\omega}}} & (6)\end{matrix}$

From this, we obtainG′(ω)=

{G(ω)} and G″(ω)=

{G(ω)}.  (7)

G′(ω) measures the medium's elastic response to shear forces, and G″(ω)measures its viscosity. They are natural probes of biofilms' responsesto potential therapeutic agents. Similarly, the dynamic viscosity,

$\begin{matrix}{{\eta(\omega)} = {\frac{1}{\omega}\sqrt{{{G^{\prime 2}(\omega)} + {G^{''2}(\omega)}},}}} & (8)\end{matrix}$provides an overall impression of a biofilm's ability to exchangematerial with its surroundings.

To establish the accuracy of our holographic microrheology system, wefirst analyze the motions of a probe particle diffusing in a Newtonianfluid. The five-minute trajectory plotted in FIG. 3A was obtained for asingle polystyrene sphere of nominal radius a=0.75 μm suspended in adensity matched solution of 25% (w/w) glycerol in water whose viscosityis expected to be 1.7 mPa s. The particle was positioned with an opticaltweezer at the midplane of a 50 μm thick sample volume to minimizehydrodynamic coupling to the glass walls and then was released toacquire data.

Fitting images to the Lorenz-Mie scattering formula yields an estimatedsingle-image precision of ∈_(x)=∈_(y)=4 nm and ∈_(x)=20 nm. Themean-square displacement for each coordinate is plotted in FIG. 3Btogether with a fit to Eq. (3). All three traces are consistent withD₀=0.1695±0.0001 μm²/s. When combined with the trajectory average of theparticle's measured radius, a=0.775±0.014 μm, this suggests an overallviscosity of η=1.67±0.01 mPa s. Given the shutter period of t_(s)=1 ms,the extrapolated offsets yield ∈_(x)=∈_(y)=8±4 nm and ∈_(z)=35±8 nm.These values are consistent with the estimated single-frame resolutionand suggest that the accuracy of the position measurement is comparableto its precision.

The availability of high-resolution axial tracking data is one of theprincipal benefits of holographic particle tracking for microrheology.Consistency among the three data sets in this case confirms themeasurements' freedom from hydrodynamic coupling to the surfaces. Moregenerally such comparisons are useful for gauging a sample's isotropyand homogeneity.

Because results from the three coordinates are in agreement, we analyzethe three-dimensional mean-squared displacement,

Δr²(t)

=Σ_(j=1) ³

Δr_(j) ²(t)

, with Eqs. (4), (6) and (8) to obtain the loss modulus, G″(ω), plottedin FIG. 4A and the dynamic viscosity, η(ω), plotted in FIG. 4B. Asexpected, the glycerol-water solution acts as a Newtonian fluid whosestorage modulus, G′(ω), is too small to resolve over the range offrequencies probed. Its viscosity, η(ω)=1.680±0.001 mPa s is thereforeindependent of frequency and agrees with values obtained from bulkmeasurements.

Having established the accuracy of the three-dimensional particletracking method and the mechanical stability of our instrument, itsefficacy is shown particle-tracking microrheology by applying it to astandard non-Newtonian sample, an aqueous solution ofhigh-molecular-weight PEO. FIG. 4C shows G′(ω) and G″(ω) obtained from asingle sphere dispersed in a 17 wt % solution of 200 kDa PEO indeionized water. As for the Newtonian fluid, consistent results areobtained in all three coordinates, so that combined results arepresented in FIGS. 4C and 4D.

The viscoelastic moduli, plotted in FIG. 4C, agree quantitatively withresults reported for similar samples under comparable conditions. Theloss modulus, G″(ω), exceeds the storage modulus, G′(ω), over the entirefrequency range, which identifies this sample as a fluid, rather than agel. The associated dynamic viscosity, plotted in FIG. 4D, decreasesmonotonically with increasing frequency, which is the signature of ashear-thinning fluid.

The data in FIG. 5 show comparable results for biofilm polysaccharides.Prior studies of biofilm's structure using other methods have revealed adegree of heterogeneity at the sub-millimeter scale that might seemunamenable to systematic physical analysis. Indeed, measurements ofmodel biofilms' macroscopic rheological properties have yieldedviscoelastic moduli differing by more than three orders of magnitude,even for nominally similar samples. These differences have beenattributed to loading, strain rate, total strain and sample preparation.

Microrheology addresses many of these concerns by probing thelocal-scale properties of unloaded samples in equilibrium. Althoughparticle-tracking microrheology has been applied to a wide range ofindustrially and biologically relevant materials, its application tobiofilms appears to be novel. Model biofilms can be prepared without thecomplication of swimming bacteria, and so lend themselves to this kindof analysis.

An example study was performed on Streptococcus mutans polysaccharidesamples which were extracted from 5-day-old biofilms of S. mutans UA159(ATCC 700610) that were grown on glass slides in the presence of 10%sucrose. Water-soluble (S) polysaccharides were extracted with MilliQwater at room temperature. The insoluble (N) polysaccharide fractionthen was extracted in 1N NaOH. Both extracts were neutralized to pH7.0±0.5 and precipitated with cold ethanol (75% v/v) at −18° C. for atleast 24 h. The resulting polysaccharide samples have a mean molecularweight of about 10 kDa with a polydispersity of at least 50 percent, andcontain trace amounts of protein. After precipitation, the samples werewashed several times with 75% (v/v) ethanol, blotdried and dissolved inwater (S) or 1N NaOH (N), at 20% (w/v) to form the gels used in themicrorheological measurements. Polystyrene probe particles weredispersed at random in the polysaccharide at this time, and particlesnear the mid-plane of the sealed sample chamber were selected formeasurement.

Results for the S fraction, plotted in FIGS. 5A and 5B, resemble thosefor the PEO solution in FIGS. 4C and 4D. The biofilm's water-solublepolysaccharides form a shear-thinning fluid roughly ten times moreviscous than water.

The data in FIG. 5C, by contrast, show that the N-type fraction forms anelastic gel with a typical storage modulus of 10 Pa. This is severalhundred times smaller than the mean value reported for bulk samples ofS. mutans polysaccharides in other conventional prior art. It isconsistent, however, with values reported at the lowest loadings whenthe substantial measurement error is taken into account. Accuratemeasurements at low loading are one of the strengths of microrheology,so that the results in FIGS. 5C and 5D are more likely to reflect thebiofilm's properties in vivo. Even at low loading, the N-type gel isstrongly shear-thinning, as indicated by its dynamic viscosity in FIG.5D. This is a desirable trait for dental biofilms because it facilitatesremoval by brushing.

These observations suggest complementary roles of the two fractions inestablishing the biofilm's mechanical and biological properties in vivo.The N-type material appears better suited to play the role of themechanical scaffold within which the biofilm's bacterial colonyestablishes its ecosystem. Ability to disrupt the N-type gel, therefore,might be considered a promising characteristic for therapeutic agents.Microrheological assays of such agents' influence on N-type extractsshould provide a simple and cost-effective screening technique. Theability of holographic microrheology to track multiple probe particlessimultaneously furthermore creates opportunities for screening multipletherapeutic agents individually and in combination as a function ofconcentration.

B. Characterizing Milk Fat Globules

In another embodiment, we have used the method of the invention toanalyze milk fat droplets from a range of commercial milk productsincluding several grades of homogenized pasteurized cow's milk, sheepmilk and goat milk. In each case, the sample was diluted by 1000:1 withdeionized water before being sealed between a microscope slide and aglass cover slip and mounted on the stage of the microscope. Given theimaging system's calibrated magnification of 101 nm/pixel, a typical640×480 pixel image, I(r), captures roughly 10 resolvable globules.

Unprocessed holograms suffer from large intensity variations due tospeckle, interference effects in the microscope's optics and scatteringby dust and other imperfections. We correct for these by normalizingI(r) with a background hologram I₀(r) obtained with no sample in thefield of view. As described in part hereinbefore, the normalizedhologram then can be fit to the prediction of Lorenz-Mie theory,

$\begin{matrix}{{\frac{I(r)}{I_{0}(r)} = {1 + {2{\alpha\Re}\left\{ {{{f_{s}\left( {r - r_{p}} \right)} \cdot \hat{ɛ}}e^{{- i}\;{kz}_{p}}} \right\}} + {\alpha^{2}{{f_{s}\left( {r - r_{p}} \right)}}^{2}}}},} & (9)\end{matrix}$where k=2πn_(m)/λ is the wavenumber of light in a medium of complexrefractive index n_(m), and where f_(s)(r−r_(p)) is the Lorenz-Miescattering function describing how light of polarization {circumflexover (∈)} is scattered by a sphere located at r_(p). In practice, theilluminating beam is not perfectly uniform, and the factor α≈1 can beused to account for variations in I₀(r_(p)), such as position-dependentvariations. Assuming the scatterer to be a uniform and homogeneousdielectric sphere illuminated by light that is linearly polarized in the{circumflex over (x)} direction,

$\begin{matrix}{{{f_{s}(r)} = {\sum\limits_{n = 1}^{\infty}\;{f_{n}\left( {{{ia}_{n}{N_{e\; 1n}^{(3)}(r)}} - {b_{n}{M_{o\; 1n}^{(3)}(r)}}} \right)}}},} & (10)\end{matrix}$where f_(n)=i^(n)(2n+1)/[n(n+1)], and where M_(oln) ⁽³⁾(r) and N_(eln)⁽³⁾(r) are well known vector spherical harmonics,

$\begin{matrix}{\mspace{79mu}{{M_{o\; 1n}^{(3)}(r)} = {{\frac{\cos\;\phi}{\sin\;\theta}P\frac{1}{n}\left( {\cos\;\theta} \right){j_{n}({kr})}\hat{\theta}} - {\sin\;\phi\frac{d\;{P_{n}^{1}\left( {\cos\;\theta} \right)}}{d\;\theta}{j_{n}({kr})}\hat{\phi}}}}} & (11) \\{{N_{{e\; 1n}\;}^{(3)}(r)} = {{{n\left( {n + 1} \right)}\cos\;\phi\;{P_{n}^{1}\left( {\cos\;\theta} \right)}\frac{j_{n}({kr})}{kr}\hat{r}} + {\cos\;\phi\frac{d\;{P_{n}^{1}\left( {\cos\;\theta} \right)}}{d\;\theta}\frac{1}{kr}{\frac{d\;}{d\; r}\left\lbrack {{rj}_{n}({kr})} \right\rbrack}\hat{\theta}} - {\frac{\sin\;\phi}{\sin\;\theta}{P_{n}^{1}\left( {\cos\;\theta} \right)}\frac{1}{kr}{\frac{d\;}{d\; r}\left\lbrack {{rj}_{n}({kr})} \right\rbrack}{\hat{\phi}.}}}} & (12)\end{matrix}$Here, P_(n) ¹ (cos θ) is the associated Legendre polynomial of the firstkind, and j_(n)(kr) is the spherical Bessel function of the first kindof order n. The expansion coefficients in Equation (10) are given byconventional well known relationship,

$\begin{matrix}{{a_{n} = \frac{{m^{2}{{j_{n}({mka})}\left\lbrack {{kaj}_{n}({ka})} \right\rbrack}^{\prime}} - {{j_{n}({ka})}\left\lbrack {{mkaj}_{n}({mka})} \right\rbrack}^{\prime}}{{m^{2}{{j_{n}({mka})}\left\lbrack {{kah}_{n}^{(1)}({ka})} \right\rbrack}^{\prime}} - {{h_{n}^{(1)}({ka})}\left\lbrack {{mkaj}_{n}({mka})} \right\rbrack}^{\prime}}}{{and}\mspace{14mu}{by}}} & (13) \\{b_{n} = \frac{{{j_{n}({mka})}\left\lbrack {{kaj}_{n}({ka})} \right\rbrack}^{\prime} - {{j_{n}({ka})}\left\lbrack {{mkaj}_{n}({mka})} \right\rbrack}^{\prime}}{{{j_{n}({mka})}\left\lbrack {{kah}_{n}^{(1)}({ka})} \right\rbrack}^{\prime} - {{h_{n}^{(1)}({ka})}\left\lbrack {{mkaj}_{n}({mka})} \right\rbrack}^{\prime}}} & (14)\end{matrix}$where m=n_(p)/n_(m) is the particle's refractive index relative to themedium, j_(n)(x) is the spherical Bessel function of the first type oforder n, h_(n) ⁽¹⁾(x) is the spherical Hankel function of the first typeof order n, and where primes denote derivatives with respect to theargument.The sum in Eq. (2) converges after a number of terms,n_(c)=ka+4.05(ka)^(1/3)+2, which depends on the particle's size. Theonly challenge in this computation is to calculate the Bessel functionsand their ratios both accurately and efficiently. Most preferably oneuses an accurate, computationally intensive continued fraction algorithmwhich is a conventional method from Lentz to compute the a_(n) and b_(n)coefficients. In addition, the more efficient recursive algorithm due toWiscombe can be used for the spherical Bessel functions. This trade-offensures that we can obtain accurate results using our apparatus forspheres ranging in diameter from 10 nm to more than 10 μm and refractiveindexes exceeding n_(p)=2.6.

To characterize a sphere, we fit its normalized hologram to Eqs. (9)through (14) for r_(p), n_(p), a, and α, using a standardLevenberg-Marquardt least-squares algorithm. Despite the fairly largenumber of free parameters, these fits converge rapidly and robustly, andtypically yield the particle's position and size with nanometer-scaleresolution, and its refractive index to within one part in a thousand.

We applied this technique to 5 samples of commercially processed milkobtained from a local supermarket. These include pasteurized homogenizedcow's milk with designated fat contents ranging from fat-free to wholemilk, as well as goat's milk. Up to 100 randomly selected fat dropletswere analyzed for each sample to obtain estimates for the size andrefractive index distributions for the fat droplets in each sample. Theresults are summarized in FIGS. 6A-6C and Table I.

TABLE I Globule radius and refractive index for commercial milk samples.Sample Radius [^(μ)m] Refractive Index Elmherst Whole 0.693 ± 0.1741.468 ± 0.035 Elmherst 2% 0.643 ± 0.183 1.460 ± 0.029 Elmherst 1% 0.590± 0.131 1.465 ± 0.034 Elmherst Fat Free 0.562 ± 0.140 1.460 ± 0.031Meyenberg Goat Fresh 0.576 ± 0.137 1.425 ± 0.038 Meyenberg Goat 1 Mo.0.441 ± 0.088 1.451 ± 0.036 Type B Immersion Oil 1.521 ± 0.017

The radius of the fat globules increases with increasing fat content, asdoes the dispersion of the radius. The mean refractive index of theindividual globules from cow's milk, n=1.464 agrees with sample-averagedvalues for single-droplet refractive indexes obtained by lightscattering. It substantially exceeds the range of 1.3444 to 1.3525obtained for the overall refractive index of bulk samples. Such valuesare dominated by the optical properties of water, whose refractive indexat room temperature is n_(m)=1.333. Holographic measurements, bycontrast, yield values for individual droplets, and so offer a moredetailed view of the fat's properties than either of these previouslyreported methods. Indeed, it is possible to distinguish goat's milk fromcow's milk on this basis, the goat's milk having a resolvably lower meanrefractive index.

Interestingly, this relationship changes as the milk ages. Table I showsthat the mean refractive index of goat milk droplets increases over thecourse of a month, rising to 1.45. This result demonstrates thatholographic characterization can be useful not only for distinguishingtypes of milk, but also can be used to assess a sample's age (which isassociated with chemical changes).

Having access to particle-resolved data also reveals an interestingcorrelation between the size of the fat globules and the dispersion oftheir estimated refractive indexes. Larger particles' apparentrefractive indexes are consistent with each other to within theresolution of the measurement technique. Smaller droplets display asubstantially larger range of refractive indexes.

This is not an inherent limitation of the measurement technique, as thedata in FIGS. 7A-7C demonstrate. The circular data points were obtainedfor droplets of Cargille Type B microscope immersion oil that weredispersed as spherical droplets in water by vigorous shearing. This oilhas a nominal bulk refractive index of 1.515 for red light at atemperature of 25±C. This value is consistent with the single-dropletresults obtained over the entire range of droplet radii considered inthis study, ranging from 0.25 μm to 2.5 μm. Variations from droplet todroplet may be ascribed to imperfect correction of intensity variationsin the illumination by Eq. (9).

Results more reminiscent of those for milk fat droplets are obtainedwhen the oil droplets are stabilized with surfactant. The square pointsin FIG. 7A were obtained for Type B oil with the addition of 0.1% (v/v)Tergitol NP9, a nonionic surfactant whose bulk refractive index is1.491. The addition of this surfactant reduces the single-dropletrefractive index for larger particles. It also increases the range ofrefractive indexes measured for smaller particles.

This observation leads us to conclude that holographic particlecharacterization is sensitive to surface properties, and, in particular,to surface coverage by surfactants. In the case of milk droplets, thissuggests that holographic microscopy is sensitive to the milk fatglobule membrane (MFGM). This sensitivity is noteworthy because, at just10 to 20 nm thickness, the MFGM is much thinner than the wavelength oflight and constitutes a very small proportion of the droplets' volume.

The variability in results obtained for smaller droplets most likelyreflects the breakdown of assumptions underlying the derivation of Eqs.10 through 14. This form of the Lorenz-Mie scattering theory isappropriate for a homogeneous isotropic sphere with an abrupt interface.Using this result to interpret holograms of coated spheres consequentlycan lead to inconsistencies in the extracted parameters. This effectshould be more pronounced for smaller spheres whosesurface-area-to-volume ratio is higher. Applying a more sophisticatedform of the scattering function that accounts for core-shell structureshould reduce this variability at the expense of considerable additionalcomputational complexity.

Smaller milk fat droplets also display systematically larger refractiveindexes than larger droplets. This may reflect size-dependent variationin the fatty acid composition of the MFGM and triglyceride core. Eventhe simplest implementation of holographic characterization thereforecan be useful for assessing milk fat droplet composition.

Still more information could be obtained by performing holographiccharacterization of individual droplets at multiple wavelengthssimultaneously. The resulting spectroscopic information could be usefulfor further quantifying the composition of individual globules. Even inits simplest form, however, fat globule characterization throughholographic microscopy provides a particle-by-particle analysis of milkfat composition that is not otherwise available. It requires littlespecialized equipment, and so can be easily adapted for process controland quality assurance applications.

The particular application of holographic characterization to milkdemonstrates that it is possible to determine both the type (cow, goat,etc.) and quality (fat-free, whole, fresh, old, etc.) of milk samplesbased on holographic analysis. This observation suggests particularapplications of holographic characterization to quality assurance andprocess control in dairy industries. More generally, the broader use ofholographic characterization for other emulsion-based systems, such aspaint, other foods, and cosmetics is clearly applicable based on thisdisclosure herein.

It should also be noted that holographic characterization is sensitiveto the surface properties of emulsion droplets, as well as to their bulkproperties. Surface characterization can include identifying theexistence of a surface coating, measurement of surface coverage, andcharacterization of the nature of surface coating. Fitting to a moresophisticated form of the theory can provide quantitative information onthe thickness and composition of surface coatings. This is useful formilk characterization, and it also should be useful in other contexts(indicated above) where the surface of a particle can differ from itsbulk.

C. Characterizing Colloidal Particles

In yet another embodiment, a polystyrene sulfate sphere dispersed inwater was analyzed and characterized. As described hereinbefore,digitized holograms yield a particle's three-dimensional position,r_(p), its radius, a, and its index of refraction, n_(p). We assume thatthe incident field, E₀(r)=u₀(p)exp(ikz){circumflex over (∈)}, isuniformly polarized in the {circumflex over (∈)} direction and variesslowly enough over the size of the particle to be treated as a planewave propagating along the {circumflex over (z)} direction. Itsamplitude u₀(ρ) at position ρ=(x, y) in the plane z=z_(p) of theparticle is thus the same as its amplitude in the focal plane, z=0. Thewave propagates along the {circumflex over (z)} direction with wavenumber k=2πn_(m)/λ, where λ, is the light's wavelength in vacuum andn_(m) is the refractive index of the medium. For pure water at 25° C.,n_(m)=1.3326 at λ=0.632 μm.

The particle at r_(p) scatters a portion of the incident field into ahighly structured outgoing wave, E_(s)(r)=αexp(−ikz_(p))u_(o)(r_(p))f_(s)(r−r_(p)), where α=1 accounts forvariations in the illumination, and where f_(s)(r) is the Lorenz-Miescattering function, which depends on a, n_(p), n_(m) and λ. Thescattered field generally covers a large enough area at the focal planethat the interference pattern,I(ρ)=|E _(s)(r)+E ₀(r)|²|_(z=0),  (15)is dominated by long-wavelength variations in |u₀(ρ)|². The resultingdistortions have been characterized, but were not corrected in previousanalyses of I(ρ). Fortunately, |u₀(ρ)|² can be measured in an emptyfield of view, and the in-line hologram can be normalized to obtain theundistorted image,

$\begin{matrix}{{B(\rho)} \equiv \frac{I(\rho)}{{{u_{0}(\rho)}}^{2}}} & (16) \\{{= {1 + \frac{2\Re\left\{ {{E_{s}(r)} \cdot {E_{0}^{*}(r)}} \right\}}{{{u_{0}(\rho)}}^{2}} + \frac{{{E_{s}(r)}}^{2}}{{{u_{0}(\rho)}}^{2}}}},} & (17)\end{matrix}$on the plane z=0. If we further assume that the phase of the collimatedincident beam varies slowly over the field of view, the normalized imageis related to the calculated Mie scattering pattern, f_(s) (r), in theplane z=0 by,B(ρ)≈1+2α

{f _(s)(r−r _(p))·{circumflex over (∈)}e ^(−ikz) ^(p) }+a ² |f _(s)(r−r_(p))|²  (18)

Eq. (18) can be fit to measured holograms by treating the particle'sthree-dimensional position, its radius and its refractive index as freeparameters. Previous studies fit non-normalized holograms tophenomenological models or Mie scattering theory for some of thesequantities, but never all five. Because errors in the adjustableparameters are strongly correlated, failing to optimize them allsimultaneously yields inaccurate results. Fitting instead to the fullLorenz-Mie theory provides more information with greater precision.

Numerical fits to digitized and normalized holographic images wereperformed with the Levenberg-Marquardt nonlinear least-squaresminimization algorithm using the camera's measured signal-to-noise ratioto estimate single-pixel errors. The χ² deviates for all of the fits wereport are of order unity, so that the calculated uncertainties in thefit parameters accurately reflect their precision.

Because the laser's wavelength and the medium's refractive index areboth known, the only instrumental calibration is the overallmagnification. This contrasts with other three-dimensional particletracking techniques, which require independent calibrations for eachtype of particle, particularly to track particles in depth.

The image in FIG. 8A(1) (and accompanying data of 8A(2) and 8A(3)) showsthe normalized hologram, B(ρ), for a polystyrene sulfate spheredispersed in water at height z_(p)=22.7 μm above the focal plane. Thissphere was obtained from a commercial sample with a nominal diameter of2a=1.48±0.03 μm (Bangs Labs, Lot PS04N/6064). The camera's electronicshutter was set for an exposure time of 0.25 msec to minimize blurringdue to Brownian motion. After normalizing the raw 8-bit digitizedimages, each pixel contains roughly 5 significant bits of information.The numerical fit to B(ρ) faithfully reproduces not just the position ofthe interference fringes, but also their magnitudes. The quality of thefit may be judged from the azimuthal average; the solid curve is anangular average about the center of B(ρ), the dashed curves indicate thestandard deviations of the average, and the discrete points are obtainedfrom the fit.

The fit value for the radius, a=0.73±0.01 μm, (see FIG. 8A(2)) falls inthe sample's specified range, which reflected a lower bound of 0.69±0.07μm obtained with a Beckman Z2 Coulter Counter and an upper bound of0.76±0.08 μm obtained by analytical centrifugation. Agreement betweenthe quoted and measured particle size suggests that the presentmeasurement's accuracy is comparable to its precision. In that case,both precision and accuracy surpass results previously obtained throughanalysis of I(ρ). The trajectory-averaged value for the refractiveindex, n_(p)=1.55±0.03, also is consistent with the properties ofpolystyrene colloid inferred from light scattering measurements on bulkdispersions.

Comparable precision in measuring a single particle's refractive indexhas been achieved by analyzing a colloidal particle's dynamics in anoptical trap. This method only can be applied to particles withcomparatively small refractive indexes, however, because particles withrelative refractive indexes greater than n_(p)≈1.3n_(m) are difficult totrap. Holographic characterization, by contrast, requires only a singleholographic snapshot rather than an extensive time series, does notrequire optical trapping, and so does not require separate calibrationof the trap, and is effective over a wider range of particle sizes andrefractive indexes.

Additional data in FIGS. 8B(1) through 8B(3) were obtained for a 1.45 μmdiameter TiO₂ sphere at z_(p)=7 μm above the focal plane. This samplewas synthesized from titanium tetraethoxide and was heat-treated toincrease its density. Strong forward scattering by such high-indexparticles gives rise to imaging artifacts unless the medium is indexmatched to the cover slip. Dispersing the particle in immersion oil(n_(m)=1.515) eliminates these artifacts, but introduces sphericalaberration for the lens we used, which must be corrected to obtainreliable results. The fit diameter of 1.45±0.03 n_(m) and refractiveindex of 2.01±0.05 are consistent with results obtained by electronmicroscopy and bulk light scattering, respectively. This result isnoteworthy because no other single-particle characterization methodworks for such high refractive indexes.

The data in FIGS. 8C(1) through 8C(3) show results for a nominally 5 μmsilica sphere (Bangs Labs, Lot SS05N/4364) dispersed in water atz_(p)=38.8 μm above the focal plane. The fit refractive index,n_(p)=1.434±0.001, is appropriate for porous silica and the diameter,a=4.15±0.01 μm agrees with the 4.82±0.59 μm value obtained for thissample with a Beckman Z2 Coulter Counter. We have successfully appliedholographic characterization to colloidal spheres as small as 100 nm indiameter and as large as 10 μm. Unlike model-based analytical methods,fitting to the exact Lorenz-Mie scattering theory is robust and reliableover a far wider range of particle sizes, provided that care is taken tomaintain numerical stability in calculating f_(s)(r).

The same fits resolve the particle's position with a precision of 1 nmin-plane and 10 nm along the optical axis. Comparable nanometer-scaletracking resolution can be obtained with conventional illumination, butrequires detailed calibrations for each particle. Another benefit ofholographic imaging is its very large depth of focus compared withconventional microscopy. Our system provides useful data over a range ofmore than 100 μm, which contrasts with the ±3 μm useful depth of focususing conventional illumination.

Holographic video microscopy lends itself to three-dimensional particletracking, as the data in FIGS. 9A(1) through 9B(2) demonstrate for acolloidal silica sphere (Bangs Labs, Lot SS04N/5252) dispersed in water.This particle was lifted 30 μm above the focal plane with an opticaltweezer, and then released and allowed to sediment. The images in FIGS.9A(1) and 9B(1) show the particle near the beginning of its trajectoryand near the end. Fits to Eq. 18 are shown in FIGS. 9A(2) and 9B(2).

The particle's measured trajectory in 1/30 s intervals during 15 s ofits descent is plotted in FIG. 10. Its vertical position z(t), FIG. 11,displays fluctuations about a uniform sedimentation speed, ν=1.021±0.005μm/s. This provides an estimate for the particle's density throughρ_(p)=ρ_(m)+9nv/(2a²g) where ρ_(m)=0.997 g/cm³ is the density of waterand η=0.0105 P is its viscosity at T=21° C., and where g=9.8 m/s² is theacceleration due to gravity. The fit value for the particle's radius, ata=0.729±0.012 μm, remained constant as the particle settled. This valueis consistent with the manufacturer's specified radius of 0.76±0.04 μm,measured with a Beckman Z2 Coulter Counter. Accordingly, we obtainρ_(p)=1.92±0.02 g/cm³, which is a few percent smaller than themanufacturer's rating for the sample. However, the fit value for therefractive index, n_(p)=1.430±0.007, also is 1.5% below the rated value,suggesting that the particle is indeed less dense than specified.

The mean-square displacements, Δr_(j) ²(τ)=

Δr_(j)(t+τ)−r_(j)(t))²

, of the components of the particle's position provide additionalconsistency checks. As the data in FIG. 12 shows, fluctuations in thetrajectory's individual Cartesian components agree with each other, andall three display linear Einstein-Smoluchowsky scaling, Δr_(j) ²(r)=2Dτ,a diffusion coefficient D=0.33±0.03 μm²/s. This is consistent with theanticipated Stokes-Einstein value, D₀=k_(B)T/(6πηa)=0.30±0.03 μm²/s,where k_(B) is Boltzmann's constant. The offsets obtained from linearfits to Δr_(j) ²(t) also are consistent with no worse than 1 nm accuracyfor in-plane positions and 10 nm for axial positions throughout thetrajectory. The optical characterization of the particle's propertiesthus is consistent with the particle's measured dynamics.

Precise measurements of probe particles' three-dimensional trajectoriesmade possible by video holographic microscopy lend themselves naturallyto applications in particle-imaging microrheology. Applying thistechnique to biofilms, in particular, shows promise for high-throughputcombinatorial screening of candidate therapeutic or remedial agents.Rather than assessing their biological or biochemical influence,holographic microrheology offers direct insight into these agents'influence on biofilms' physical properties. In the case of dentalbiofilms, the availability of model polysaccharide gels will greatlysimplify the development of standard assays for therapeutic agents.Because microrheological measurements require only micrometer-scalesamples, very large arrays of independent assays should be possible incentimeter-scale systems, with each assay requiring just a few minutesof holographic recording.

The techniques we have described are readily extended for particles andmedia whose refractive indexes have large imaginary components.Extensions for core-shell particles and particles with more complexshapes, such as cylindrical nanowires, similarly should be feasible.

We have demonstrated that a single snapshot from an in-line holographicmicroscope can be used to measure a colloidal sphere's position and sizewith nanometer-scale resolution, and its refractive index with precisiontypically surpassing 1 percent.

A video stream of such images therefore constitutes a powerfulsix-dimensional microscopy for soft-matter and biological systems.Holographic particle tracking is ideal for three-dimensionalmicrorheology, for measuring colloidal interactions and as force probesfor biophysics. The methods we have described can be applied to trackinglarge numbers of particles in the field of view simultaneously forhighly parallel measurements. Real-time single-particle characterizationand tracking of large particle ensembles will be invaluable in suchapplications as holographic assembly of photonic devices. Applied tomore highly structured samples such as biological cells and colloidalheterostructures, they could be used as a basis for cytometric analysisor combinatorial synthesis.

In addition, the concept of multi-wavelength holographiccharacterization can be applied to obtain spectroscopic information,such as the dependence of refractive index on wavelength. This can beuseful for quantifying nutrient concentration in milk and relatedsystems. It can be useful for distinguishing adulterants from pureproducts. Multiwavelength characterization also provides an opportunityfor calibration-free measurements. The concept is that the particle'smeasured size should be independent of laser wavelength. If thewavelengths of the lasers used to illuminate the sample are known, thenthe overall length-scale calibration can be obtained form the conditionthat the particle's apparent radius should be the same at allwavelengths.

While preferred embodiments have been illustrated and described, itshould be understood that changes and modifications can be made thereinin accordance with one of ordinary skill in the art without departingfrom the invention in its broader aspects. Various features of theinvention are defined in the following claims.

What is claimed:
 1. A method for characterizing a specimen, comprising:providing a holographic microscope; selecting multiple wavelengths for alaser; scattering the laser's beam off the specimen to generate ascattered portion; generating an interference pattern from anunscattered portion of the collimated laser beam and the scatteredportion; recording the interference pattern for subsequent analysis;applying a Lorenz-Mie scattering function to calculate a hologram andfitting the recorded interference pattern to the calculated hologram:and determining an estimate of the specimen's refractive index andradius from the fitted calculated hologram; wherein the determining ofthe refractive index of the specimen comprises measuring refractiveindices of the specimen at each of the multiple wavelengths.
 2. Themethod of claim 1, further comprising detecting the composition of acoating on the specimen by the measured refractive indices.
 3. Themethod of claim 1, performing a multi-wavelength characterization forcalibration free measurements by using different but known wavelengthsto obtain a length-scale calibration.
 4. The method of claim 1, whereinselecting characteristics comprises selecting a pulsed nature andfurther comprising altering one of shape or timing of the pulsed laser.5. The method of claim 4, further comprising synchronizing timing of thepulsed nature with a camera shutter associated with the holographicmicroscope.
 6. The method of claim 1, further comprising obtainingtrajectory data for the specimen.
 7. The method of claim 1, furthercomprising analyzing the trajectory data to determine one of fieldvelocity and viscoelastic properties.
 8. The method as defined in claim1 further comprising determining an estimate of the specimen's position.9. The method as defined in claim 1 wherein the holographic microscopeincludes a focal plane and the step of measuring comprises identifyingaxial displacement of the specimen relative to the focal plane whereinthe specimen comprises a particle disposed within a medium.
 10. Themethod of claim 9, further comprising selecting a depth of focus of morethan 100 μm.
 11. A computer implemented system comprising: a holographicmicroscope apparatus comprising a coherent light source with multiplediscrete concurrent wavelengths of coherent light beams' a specimenstage, an objective lens, and an image collection device; a computermodule in communication with the holographic microscope apparatus andincluding a processor and memory, the memory receiving image data fromthe image collection device and further having a set of instructionsfor; selecting multiple wavelengths for a laser; scattering the laser'sbeam off the specimen to generate a scattered portion; generating aninterference pattern from an unscattered portion of the collimated laserbeam and the scattered portion; recording the interference pattern forsubsequent analysis; applying a scattering function to analyze therecorded interference pattern wherein the scattering function comprisesa Lorenz-Mie function; normalizing the interference pattern by dividingthe interference pattern by a background form of interference pattern;fitting a calculated hologram to the interference pattern; anddetermining an estimate of the specimen's refractive index and radiusfrom the fitted calculated hologram; wherein the determining of therefractive index of the specimen comprises measuring refractive indicesof the specimen at each of the multiple wavelengths.
 12. The computerimplemented system of claim 11, where the computer module furtherincludes instructions for detecting the composition of a coating on thespecimen by the measured refractive indices.
 13. The computerimplemented system of claim 11, where the computer module furtherincludes instructions for performing a multi-wavelength characterizationfor calibration free measurements by using different but knownwavelengths wavelengths to obtain a length-scale calibration.
 14. Thecomputer implemented system as defined in claim 11 wherein the computermodule further includes instructions for: performing the measuring by:at least one of (1) identifying in one single time snapshot a positionof a particle in the specimen and characterizing properties of theparticle, thereby generating particle data; and (2) identifying in aplurality of time snapshots a trajectory of a particle in the specimenand characterizing properties along the trajectory, thereby generatingparticle data; and (3) wherein the specimen comprises a medium holding acolloidal suspension of particles and the step of measuring comprisesanalyzing at least one of the medium and interaction between theparticles of the suspension.